Question about the gaps between prime numbers

  • #1
222
37
Is there any prime number pn, such that it has a relationship with the next prime number pn+1
[tex] p_{n+1} > p_{n}^2 [/tex]
If not, is there any proof saying a prime like this does not exist?

I have the exact same question about this relation:
[tex] p_{n+1} > 2p_{n} [/tex]
 

Answers and Replies

  • #2
14,394
11,709
Is there any prime number pn, such that it has a relationship with the next prime number pn+1
[tex] p_{n+1} > p_{n}^2 [/tex]
If not, is there any proof saying a prime like this does not exist?

I have the exact same question about this relation:
[tex] p_{n+1} > 2p_{n} [/tex]
https://en.wikipedia.org/wiki/Prime_gap

There is also a proof for arbitrary gaps, but see the section "upper bounds".
 
  • #4
222
37
Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
[tex] p_{n+1} > p_{n}^2 [/tex]

I wonder if this can be proved or disproved from other postulates...
 
  • #5
22,089
3,296
Interesting.Bertrand's Postulate answers the second part of my question. :)

I see Firoozbakht's conjecture, which is similar to my first part, but it's not quite the same thing as
[tex] p_{n+1} > p_{n}^2 [/tex]

I wonder if this can be proved or disproved from other postulates...
This also follows very easily from Bertrand's postulate.
 
  • #6
Stephen Tashi
Science Advisor
7,583
1,472
but it's not quite the same thing as
[tex] p_{n+1} > p_{n}^2 [/tex]

I wonder if this can be proved or disproved from other postulates...
Compare ##2p_n## to ##p^2_n## .
 
  • #7
222
37
This also follows very easily from Bertrand's postulate.
Yeah it does. Wow. I'm dumb. :p
 

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